Hierarchies based on computational complexity and irregularities of class determining measured sets (Preliminary Report)

Abstract

We consider here the problem of building transfinite hierarchies of computable functions on the basis of their difficulty of computation. Previous hierarchies of functions through the constructive ordinals have had two major problems, each apparently caused by not having techniques to restrict the classes considered at limit ordinals. These problems are, first that every function occurs at some name for o, or some other small ordinal, and second that two names for the same constructive ordinal have two different classes of functions associated with them. The axioms of computational complexity introduced by Blum[1] and several theorems by Blum[1],and by McCreight and Meyer[9], particularly the union theorem, lead to a very natural method of hierarchy building that partially avoids the first of these problems. The other problem is not overcome, however. In attempting to understand the problem of obtaining unique, or nearly unique classes of functions we have encountered difficulties because the class determining measured sets of McCreight and Meyer[9] lack several properties that might reasonably be desired.